Integral dari $$$\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}$$$

Temukan $$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Tulis ulang integran:

$${\color{red}{\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{2 \cos{\left(x \right)} d x}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=2$$$ dan $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$${\color{red}{\int{2 \cos{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{\cos{\left(x \right)} d x}\right)}}$$

Integral dari kosinus adalah $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$2 {\color{red}{\int{\cos{\left(x \right)} d x}}} = 2 {\color{red}{\sin{\left(x \right)}}}$$

Oleh karena itu,

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}$$

Tambahkan konstanta integrasi:

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}+C$$

$$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 2 \sin{\left(x \right)} + C$$$A